Eigenvalue and Stability Problems in Applied Mathematics

该奖项将支持的研究涉及应用数学中具有特征值问题的共同主题的一些主题。本征值问题出现在科学各个分支的许多背景下，特别是在与物理系统稳定性有关的问题中。具体地说，在数学物理中出现的许多方程中，可能存在不稳定的有效精确解，即附近的解随着时间的推移而迅速偏离该精确解。这样的解通常很难或不可能在实验中实现，因为必须非常精确地选择初始条件，才能使解在任何可察觉的时间长度内保持可观测。换句话说，物理上有趣的解决方案通常是稳定的。什么不是。不幸的是，通常很难确定某一特定溶液是否稳定。根据该奖项，将研究的问题范围从光子材料中的水波和波传播到生理学模型。具体地说，将研究与缺陷光子晶体有关的问题，非线性薛定谔方程的驻波问题，玻色-爱因斯坦凝聚的涡旋晶体问题，与Sine-Gordon方程有关的散射问题，以及神经生理学(大脑的动眼神经积分器)问题。在其中一些项目中，将开发几何方法来确定解的不稳定性。这些方法非常普遍，它们基于几何考虑，而不是所讨论的方程的细节。因此，它们可能适用于科学中许多学科的问题。还将使用其他技术，如渐近性和拓扑学论证。稳定性是许多物理系统的基本属性。例如，挂在链子上的怀表是一个稳定的系统--当受到干扰时，它最终会恢复到静止状态。一支铅笔顶端立在桌子上是不稳定的--一个很小的扰动就会导致它倒下。然而，对于许多工程化的物理系统来说，它是否稳定尚不清楚。例子出现在新型光学材料的设计中，其中行进光脉冲的稳定性是令人感兴趣的。在其他情况下，例如在生理学上，不稳定与故障有关，探索导致不稳定的机制是有意义的。例如，动眼运动积分器是移动眼睛的大脑子系统的一部分，只要数学模型满足适当的条件(一个主要的特征值在原点附近)，就知道它可以正常工作。如果在不应该存在特征值的位置存在特征值，系统就会变得不稳定，从而导致一种被称为先天性眼球震颤的疾病，大约每2000人中就有一人受到影响。这个项目的研究主要涉及最广泛意义上的稳定性问题。它将提供适用于从光子学到生理学的各种问题的工具。

The research that will be supported by this award addresses a number of topics in applied mathematics that have the common theme of eigenvalue problems. Eigenvalue problems arise in a number of contexts in various branches of science, in particular in questions related to the stability of a physical system. Specifically, in many equations arising in mathematical physics there may exist valid exact solutions which are unstable, in the sense that nearby solutions rapidly diverge from this exact solutions as time progresses. Such solutions are often difficult or impossible to realize in an experiment, since the initial conditions must be chosen very precisely in order for the solutions to remain observable for any appreciable length of time. Put differently, physically interesting solutions typically are the stable ones. and what is not. Unfortunately it is often very difficult to determine whether a particular solution is stable or not. The problems that will be studied under this award range from water waves and wave propagation in photonic materials to models in physiology. Specifically, problems related to photonic crystals with defects, standing waves for nonlinear Schroedinger equations, vortex crystals for Bose-Einstein condensates, scattering problems related to the Sine-Gordon equations, and a problem from neurophysiology (oculomotor integrator of the brain) will studied. In some of these projects, geometric methods for establishing the instability of solutions will be developed. These methods are very general, being based on geometric considerations rather than details of the equation in question. Thus they are potentially applicable to problems in many disciplines within science. Other techniques such as asymptotics and topological arguments will also be employed. Stability is a fundamental property of many physical systems. For example, a pocket watch hanging from a chain is a stable system - it will eventually return to its rest state when perturbed. A pencil that is balancing on its tip on a table is unstable - a very small perturbation will cause it to fall over. For many engineered physical systems, it is however not clear if it is stable or not. Examples occur in the design of novel optical materials, where the stability of a traveling light pulse is of interest. In other situations, e.g. in physiology, instability is related to malfunction, and it is of interest to explore mechanisms that lead to instability. For example, the oculomotor integrator, which is part of the brain subsystem that moves the eyes, is known to function properly as long as a mathematical model satisfies a suitable condition (a single dominant eigenvalue is near the origin). The system becomes unstable if there are eigenvalues in locations where they not are supposed to be, resulting in a condition known as congenital nystagmus that effects one in about 2000 people. The research in this project is primarily concerned with questions of stability in the broadest sense. It will provide tools that are applicable to problems ranging from photonics to physiology.